University of Nebraska - Lincoln
DigitalCommons@University of Nebraska - Lincoln
Ilya Fabrikant Publications
Research Papers in Physics and Astronomy
2016
Dissociative electron attachment to halogen
molecules: Angular distributions and nonlocal
effects
Ilya I. Fabrikant
University of Nebraska-Lincoln, [email protected]
Follow this and additional works at: http://digitalcommons.unl.edu/physicsfabrikant
Fabrikant, Ilya I., "Dissociative electron attachment to halogen molecules: Angular distributions and nonlocal effects" (2016). Ilya
Fabrikant Publications. 8.
http://digitalcommons.unl.edu/physicsfabrikant/8
This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska Lincoln. It has been accepted for inclusion in Ilya Fabrikant Publications by an authorized administrator of DigitalCommons@University of Nebraska Lincoln.
PHYSICAL REVIEW A 94, 052707 (2016)
Dissociative electron attachment to halogen molecules: Angular distributions and nonlocal effects
I. I. Fabrikant
Department of Physics and Astronomy, University of Nebraska, Lincoln, Nebraska 68588-0299, USA
(Received 23 August 2016; published 28 November 2016)
We study dissociative electron attachment (DEA) to the ClF and F2 molecules. We formulate a method for
calculation of partial resonance widths and calculate the angular distributions of the products in the ClF case
using the local and nonlocal versions of the complex potential theory of DEA. They show the dominance of
the p wave except in a narrow energy region close to zero energy. Comparison of the local and nonlocal DEA
cross sections show that the former are smaller than the latter by a factor of 2 in the energy region important
for calculation of thermal rate coefficients. This result is confirmed by comparison of the local and nonlocal
calculations for F2 . Only at low energies below 30 meV the local cross sections exceed nonlocal due to the 1/E
divergence of the local results. On the other hand, the thermal rate coefficients generated from the local cross
sections agree better with experiment than those calculated from the nonlocal cross sections. The most likely
reason for this disagreement is the overestimated resonance width in the region of internuclear distances close to
the point of crossing between the neutral and anion potential-energy curves.
DOI: 10.1103/PhysRevA.94.052707
I. INTRODUCTION
Comparative studies of dissociative electron attachment
(DEA) to halogen molecules is interesting for several reasons.
Due to high electron affinity of halogen atoms these reactions
are exothermic and occur with relatively high rates. Lowenergy DEA to homonuclear halogen molecules, like F2 and
Cl2 , is dominated by the u resonance with the p wave as a
main component [1,2]. This leads, according to the Wigner
threshold law for exothermic reactions [3], to E 1/2 behavior
of the cross section as a function of electron energy E at low
values of E. Confirmation of this prediction became possible
after development of laser electron attachment method [4]
generating very highly resolved in energy electron beams.
The Wigner threshold law was observed for Cl2 [5,6] and
F2 [7]. Another confirmation was obtained by comparison of
calculated and measured thermal rate coefficients for these
halogen compounds [8,9]. The theoretical calculations were
carried out within the framework of the nonlocal complex
potential theory [10] and equivalent resonance R-matrix theory
[11]. This is essential for the threshold behavior since the local
theory generates cross sections diverging as 1/E at E → 0.
There are no systematic studies of local versus nonlocal
dynamics for DEA processes. It is known that the local theory
for DEA fails in the case of a broad shape resonance with a
typical example of low-energy DEA to the H2 molecule [10,12]
driven by a very broad 2 u resonance. A similar situation is
observed in the case of hydrogen halides [13–15] where, in
addition to the effect of the broad resonance, the long-range
dipolar interaction strongly affects the DEA dynamics which
cannot be described by the local theory. On the other hand,
in the case of a relatively narrow resonance at not very low
energies and in the absence of a strong long-range interaction,
the local theory was demonstrated to be very successful.
Typical examples are CF3 Cl molecule [16] and acetylene [17].
Halogen molecules do not belong to any of these classes.
On one hand, the long-range electron-molecule interaction is
relatively weak, even in the case of interhalogen compounds.
On the other hand, because the lowest anion state crosses
the neutral curve very close to the equilibrium internuclear
2469-9926/2016/94(5)/052707(9)
separation, approximations involved in the local theory become invalid. In particular the local theory does not reproduce
the correct threshold behavior of the DEA cross sections for
F2 and Cl2 molecules, as discussed above.
It is therefore important to compare results of the nonlocal
and local calculations performed with equivalent input parameters, the potential-energy curves and capture amplitudes.
It is not clear what happens with regard to validity of the
local theory when we turn to interhalogen molecules, like
ClF. The inversion symmetry is broken in this case, and
the s-wave contribution becomes essential. This is important
for two reasons. First, the s-wave admixture changes the
angular distribution of the products in the resonance scattering,
particularly in the DEA process. Second, the widely used
approximation for the capture amplitude V [18]
V =
2π
1/2
,
(1)
where is the total resonance width, strictly speaking, is
no longer valid, although it might be still reasonable for the
calculation of the total (rather than angle-differential) cross
section.
Whereas calculation of the total resonance width has
become a standard procedure based on the Breit-Wigner
parametrization of the eigenphase sums [19], calculation
of the partial widths, resolved in angular momentum l,
present a challenge. The partial width is defined within the
framework of the Feshbach projection operator theory [20]
or the resonance R-matrix theory [21]. Ab initio calculations
using these theories are very challenging. Standard scattering
codes generate the scattering matrix, and practically it would
be more efficient to extract the partial width from scattering
matrices, as it is usually done for the total width.
The purpose of this paper is twofold. First, we develop a
method for calculation of partial widths based on the paper of
Macek [22] and apply it to resonance e−ClF scattering. Our
method is somewhat similar to that used by Haxton et al. [23]
for calculation of angular distribution of the products in DEA
to the water molecule, but is formulated in such a way that it
052707-1
©2016 American Physical Society
I. I. FABRIKANT
PHYSICAL REVIEW A 94, 052707 (2016)
is easily extendable to the nonlocal treatment of the nuclear
motion. Then we compare local and nonlocal versions of the
DEA theory for the ClF molecule. The results turn out to be
somewhat surprising, and we continue a similar investigation
for electron attachment to F2 , the process relatively well
studied in the past. We find that, in spite of the singular
behavior exhibited by local cross sections at E → 0, they
produce thermal rate coefficients which are about a factor of 2
smaller than the nonlocal results.
II. LOCAL THEORY
A. S matrix parametrization
According to the standard theory [24] the S matrix for
resonance scattering can be represented as
γ ×γ
,
(2)
S = S0 − i
E − ER + i/2
where S 0 is the background S matrix, E is the projectile energy,
ER is the resonance energy, is the total width, γ is the
column of complex partial amplitudes
γl , related to the partial
√
capture amplitudes Vl as γl = 2π Vl , l = 0,1, . . . ,lmax where
lmax is the maximum number of partial waves included in
calculations, and symbol × indicates the direct product. The
partial resonance width i is determined as
l = |γl |2 .
Because of the unitarity and time-reversal symmetry constraints we also have
S0γ ∗ = γ .
(3)
To incorporate this constraint automatically, we diagonalize
first the background scattering matrix
0
e2iδ = U T S 0 U,
where δ 0 is the diagonal matrix of background eigenphases,
and U is a real orthogonal matrix diagonalizing S 0 (T indicates
the transposition). In practice it is obtained by diagonalization
of the K 0 matrix. Then γ can be written as [22]
0
γ = U eiδ x,
FIG. 1. Original (solid lines) and fitted (dashed lines) matrix
element S11 for R = 2.9 a.u.
(l = 0 through 4) have been included in calculations, only the
first three give a noticeable contribution to the width. In Fig. 1
we present the fitting results for the matrix element S11 at
R = 2.9 a.u.
In Fig. 2 we present the first two parameters γl , and in Table I
background phase shifts for l = 0, 1, and 2. Overall we see a
decrease of |γ | with R, although a small instability at higher
l appears. It is interesting that calculation of |γ |2 shows that
the p wave dominates in the resonance scattering, although it
is not apparent from the initial K matrices. This shows that, in
spite of the broken inversion symmetry, resonance scattering
in ClF is similar to Cl2 and F2 . However, at low energies the
s-wave contribution becomes dominant, and this changes the
threshold law for DEA [25].
(4)
where x is a column of real numbers. This equation is
equivalent to Eq. (3).
The parametrization procedure is reduced to the search
for n(n + 1)/2 parameters defining the S 0 matrix, where
n = lmax + 1 is the number of channels, and n parameters
defining the column x which satisfy the constraint
xi2 = .
i
In this local-theory version we neglect the energy dependencies
of γ and S 0 by choosing narrow enough energy interval.
B. Results for ClF
The described procedure was applied to K-matrices calculated for e−ClF scattering for five internuclear separation
R [25] with the use of the UCL R-matrix code [26–28]. The
fitting interval was chosen with the center at the resonance
position and the range about 2. Although five partial waves
052707-2
FIG. 2. Parameters γl for l = 0,1.
DISSOCIATIVE ELECTRON ATTACHMENT TO HALOGEN . . .
TABLE I. Background eigenphases for e-ClF scattering.
R
2.80
2.85
2.90
2.95
3.00
l=0
l=1
l=2
−0.6489
−0.5774
−0.5108
−0.4588
−0.2595
−0.3036
−0.2475
−0.1841
−0.027 15
−0.060 52
− 0.010 08
− 0.034 52
− 0.040 82
− 0.027 10
0.005 74
C. Expansion of the capture amplitude in partial waves
For calculation of the DEA process, the capture amplitude
should be expanded in partial waves and connected with the
complex parameters γl . The exact phase factor in this relation
is often ignored because of the approximate expression for
the amplitude, Eq. (1). This relation, including the phase
factor, was given in Ref. [23], but for the local version of
the DEA theory. We present here some details applicable for
the diatomic case which can be easily extended to the nonlocal
version.
According to the Feshbach theory [20], the transition
amplitude T incorporating resonance scattering can be written
as (the standard scattering amplitude is given by f = T /k)
T = Tp +
Vout Vin
,
E − ER + i/2
(5)
where Tp is the background transition amplitude, and the Vin
and Vout amplitudes are
†
Vin (ki ) = V0 v (+) ,
Vout (kf ) = v (−) V0 ,
where V0 is a unirow matrix of coupling potentials, is a unicolumnar matrix representing the wave function of
the resonance state, v (+) is the initial wave function of the
projectile with outgoing-wave boundary condition, and v (−)
is the final-state function corresponding to the ingoing wave
boundary conditions.
In our case the partial-wave expansion of v (+) has the form
2π i l vllm (r) ∗
v (+) =
i
(6)
Yl m (k̂)Ylm (r̂),
k ll r
where vllm is a matrix (assume cylindrical symmetry for
simplicity, so m is fixed) of radial wave functions which can
be written as
v = I − OS,
where I and O are matrices of ingoing-wave and outgoingwave radial solutions.
Accordingly we can expand the “in” amplitude in partial
waves as
γl
∗
i l √ Ylm
(k̂i ).
(7)
Vin (ki ) =
2π
l
Using the relation between the ingoing wave and outgoing
wave solutions [29], we also obtain
γl
Vout (kf ) =
i −l √ Ylm (k̂f ).
(8)
2π
l
Expansion (7) is written in the body frame with the polar
axis along the internuclear axis. It can be rewritten in terms
PHYSICAL REVIEW A 94, 052707 (2016)
of the angular coordinates R̂ characterizing direction of the
molecular axis with respect to the direction of the momentum
k [30]
γl
∗
i l √ Ylm
(R̂).
Vin (ki ) = (−1)m
2π
l
Parameters γl can be determined from the appropriately
normalized solution (6). However, this is unnecessary for our
purposes since it can be proven that they are identical to those
entering Eq. (2). Expand first the transition amplitude in partial
waves [31]
i l −l Tll Yl∗ m (ki )Ylm (kf ),
T = 2π i
ll m
where Tll is the transition matrix in the angular-momentum
representation. (We use the definition T = 1 − S.) Comparing
this expansion with Eqs. (5), (7), and (8), we find
γl γl Tllres
=i
E − ER + i/2
that is consistent with Eq. (2).
For DEA calculations only the entrance projectile amplitude is necessary. According to the general theory [32], the
wave function ψv (R) of the outgoing fragments can be written
as
ξvl (R)
Ylm (R̂),
ψv (r) =
R
l
where subscript v indicates the initial vibrational state. The
radial function ξvl (R) satisfies the equation
1 d2
−
+
U
(R)
−
i(R)/2
−
E
ξvl (R)
2M dR 2
γl (R)
= −i l √ ζv (R),
2π
(9)
where U (R) is the potential-energy function for the anion,
and ζv (R) is the vibrational wave function for the initial
state. Equation (9) assumes the axial recoil approximation [30]
which works well for diatomic molecules if the collision time
is short compared to the rotational period. The solution of this
equation, satisfying the outgoing-wave boundary condition, is
γl (R )
l
(10)
G(R,R ) √
ζv (R )dR ,
ξvl (R) = −i
2π
where G(R,R ) is the Green’s function of the left-hand side
of Eq. (9) corresponding to the outgoing-wave boundary
condition. The differential cross section is given by
2
2π 2 K
dσ
= 2
lim ξl (R)Ylm (R̂) .
(11)
R→∞
d
k M
l
This expression can be simplified using the Franck-Condon
(FC) principle. Equation (10) can be rewritten as [33]
γl (Rv )
ξvl (R) = −i l √
(12)
G(R,R )ζv (R )dR ,
2π
where the FC point Rv is determined from the equation
052707-3
E − W (Rv ) = v − V0 (Rv ),
(13)
I. I. FABRIKANT
PHYSICAL REVIEW A 94, 052707 (2016)
where V0 (R) is the potential-energy curve for the neutral
molecule, W (R) = U (R) − i(R)/2 is the complex potential
entering Eq. (9), and v is the energy of the initial vibrational
state. In this approximation the l dependence of the DEA
amplitude is given by γl (Rv ); therefore, the differential cross
section is given by the expression
2
dσ
l
∗
= av i γl (Rv )Ylm (R̂) ,
(14)
d
l
where the l-independent parameter av can be expressed in
terms of the total cross section σv as
σv
av =
.
(Rv )
Note that according to Eq. (13) the FC point is complex.
This requires an analytical continuation of the amplitudes
γl (R) into the complex R plane. However, our γl (R) is not
determined accurately enough for this purpose. Assuming that
the resonance is narrow enough, we can solve Eq. (13) with
ReW (R) = U (R). For wide resonances the local theory fails
anyway.
D. Results for cross sections
The results for the integrated DEA cross sections calculated
according to Eqs. (9) do not differ substantially from those
obtained in Ref. [25] based on the approximation for the
capture amplitude, Eq. (1). This is demonstrated in Fig. 3.
The two curves become distinguishable only above E = 0.5
eV where the cross section is relatively small. This difference
does not affect the thermal rate coefficients calculated in
Ref. [25].
It is known, though, that the local theory violates the
Wigner threshold law behavior at low energies. Sometimes
this deficiency is corrected by multiplying the attachment
FIG. 3. Local calculations of DEA cross sections for ClF in the
ground and first excited vibrational states. Solid (black) curves: with
approximation (1). Dashed (red) curves: without approximation (1).
Bardsley correction is not included.
FIG. 4. Differential DEA cross sections for selected energies
indicated in eV. At E = 0.01 and 0.1 eV the cross sections calculated
with the use of the FC approximation are also shown by dashed (red)
curves.
amplitude by a factor introduced by Bardsley [34]
c = (E/Er (R))a/2 ,
(15)
where Er (R) = U (R) − V0 (R) is the resonance energy at
a given R and a is the threshold exponent. For nonpolar
molecules, according to the Wigner law a = lmin + 1/2. For
the l-dependent amplitude the correction factor should become
l dependent as well, and the threshold exponent becomes al =
l + 1/2. It is known, however, that the Bardsley correction
can underestimate DEA cross sections that were indeed
observed in Ref. [25]. Moreover, our calculations have shown
that the choice al = l + 1/2 produces cross sections which
are substantially smaller than those calculated in Ref. [25];
therefore, the E 1/2 scaling seems to be more reasonable.
While the complete answer to the question about the correction
factor requires nonlocal calculations, it is safe to conclude at
this point that the present calculations of the integrated cross
sections confirm the results of Ref. [25].
In Fig. 4 we present differential cross sections for selected
energies. We also check the validity of the FC approximation
by presenting the results of local calculations obtained by
direct calculation of the integral (10) and by the use of the
FC approximation, Eq. (12). The FC approximation works
very well except at low energies below 0.1 eV. In this region
the nuclear motion is very slow and the FC principle is
less valid. However, the FC approximation affects very little
the integrated cross sections even in the low-energy region.
These observations are important for interpretation of nonlocal
results discussed below.
It is apparent that the l = 1 component is dominant
even at relatively low energies, although theoretically with
decreasing energies the s-wave component should take over.
The local theory without the Bardsley correction is not able
to describe this transition. In order to do this within the local
formalism we have to introduce the l-dependent coefficient,
Eq. (15), with al = l + 1/2. However, this modification
052707-4
DISSOCIATIVE ELECTRON ATTACHMENT TO HALOGEN . . .
PHYSICAL REVIEW A 94, 052707 (2016)
strongly reduces the total cross section and seems to be
inadequate.
TABLE II. ClF: parameters for the nonlocal width entering
Eqs. (16)–(18) in a.u.
R
III. NONLOCAL THEORY
A. Basic formalism
Nonlocal theory requires calculation of the width function
(R,E) depending both on the internuclear separation and
electron energy. Whereas calculation of the local width for a
sufficiently narrow resonance is straightforward, the nonlocal
width contains more information than the on-shell scattering
matrix. This can be seen from the expression for the Feshbach
width
†
(E) = 2π |V0 vE(+) |2 ,
which involves the scattering wave function vE(+) rather than
the scattering matrix.
A completely ab initio calculation of (R,E) is a very
challenging task even for simple diatomics and was done only
in a few cases. It is more customary in practice to parametrize
the width as a function of E and fit the parameters to reproduce
ab initio scattering matrices [10]. Although this procedure is
somewhat ambiguous, it usually gives a satisfactory way to
include the nonlocal effects in the DEA theory. The function
(E) for a given R should have a proper threshold behavior
and decay at E → ∞. A simple formula satisfying these
requirements is [10]
(E) = Ax a e−x , x = E/B,
(16)
where A and B are fit parameters and a is the threshold
exponent. In the absence of the dipolar interaction a =
lmin + 1/2 where lmin is the lowest electron angular momentum
allowed by symmetry.
The corresponding parametrization of the parameters γl (E)
can be chosen as
γl (E) = gl e−x/2 x al /2 .
(17)
2.80
2.85
2.90
2.95
3.00
B
s
0.017 115
0.012 883
0.008 412
0.004 810 6
0.001 738
1.2909
1.3253
1.4142
1.4487
1.5744
The capture amplitude of the nonlocal theory is given by
Eq. (7) with the only difference that the parameter γl is now
energy dependent. The partial wave function describing the
motion of outgoing fragments is given by the equation similar
to Eq. (9)
−
1 d2
γl (R)
+
U
(R)
+
F
−
E
ξl (R) = −i l √ ζv (R).
2M dR 2
2π
(19)
The major difference with the local theory is that it contains
a nonlocal energy-dependent operator F whose explicit expression was given by Bardsley [32] and, in more detail,
by Domcke [10]. Kalin and Kazansky [35] worked out a
quasiclassical representation for this operator, based on the
FC principle, which simplifies the solution of the nonlocal
problem. The same method is used in the present paper.
The FC principle can be also used for evaluation of the
angular distribution of the products. The result is given by the
same Eq. (14) of the local theory, except that γl (E,Rv ) are
now nonlocal energy-dependent amplitudes with the correct
threshold behavior. Note that in the nonlocal theory we are
dealing with the real diabatic anion curve U (R); therefore, the
FC point is always real, and no additional approximations in
solving Eq. (13) are involved.
Because of the small value of the dipole moment of the ClF
molecule we have chosen
al = l + 1/2.
This approximation is reasonable, since inclusion of nonzero
value of the dipole moment in the local calculations produced
negligible changes in the cross sections. To reduce the
ambiguity in fitting, we have also assumed that l dependence
of the parameter gl is the same as in the local theory; therefore,
gl = sγllocal ,
(18)
where s is an l-independent parameter. Therefore, the fitting
at each value of the internuclear distance R is reduced to
determination of s and B. The resulting fit parameters are
presented in Table II.
To increase somewhat flexibility of the fit procedure, we
have also introduced l-dependent parameter B. As will be
shown below, the results for the DEA cross sections, obtained
from the two sets of corresponding parameters, do not differ
significantly.
In Fig. 5 we present the results for (E) obtained from the
fit for several internuclear separations.
FIG. 5. Width for ClF plotted as a function of energy E for several
internuclear distances indicated in a.u.
052707-5
I. I. FABRIKANT
PHYSICAL REVIEW A 94, 052707 (2016)
FIG. 6. DEA cross sections for ClF in the ground vibrational
state. Nonlocal calculations: solid (black) curve, with l-dependent
parameter B; dashed (red) curve, with l-independent parameter B.
Local calculations: dotted (blue) curve, without Bardsley correction;
dot-dashed (magenta) curve, with Bardsley correction.
B. Comparison of nonlocal and local theories
In Fig. 6 we present comparison of four sets of the
cross sections: results of local calculations without Bardsley
correction, local calculations with the Bardsley correction, and
the nonlocal calculations with l-independent and l-dependent
parameter B in Eq. (16). The last two results are almost
indistinguishable. However, the local results, even without the
Bardsley correction, lie below the nonlocal calculations at low
energies. Only below 0.01 eV, due to the 1/E behavior, they
exceed the nonlocal results. The cross section calculated with
Bardsley correction exhibit the correct threshold behavior, but
lie well below the nonlocal results. On the other hand, at higher
electron energies (E > 0.4 eV) the local results become higher
than nonlocal.
In Fig. 7 we present angular distributions. As expected,
the nonlocal theory shows the transition from the p-wave
dominated differential cross section to the s-wave dominated.
However, the transition occurs at a very low energy region
about 0.01 eV. Therefore, DEA to interhalogens, in our case
to the ClF molecule, preserves the properties of DEA to
homonuclear halogens down to low energies. We should also
note that our nonlocal results are obtained with the use of
the FC principle; therefore, quantitatively they become less
accurate at lower energies, although the transition from the
p-wave dominance to the s-wave dominance is described
qualitatively correctly. These results might be interesting in
connection with recent observation of unexpected angular
distribution in DEA to Cl2 [36].
C. Nonlocal theory: Comparison with F2
Comparison of local with nonlocal results for the total DEA
cross sections for ClF looks surprising. It is even somewhat
disappointing because the nonlocal cross sections generate
thermal rate coefficients which are about a factor of 2 higher
FIG. 7. Differential DEA cross sections for ClF in its ground
vibrational state for selected energies calculated by the nonlocal
theory.
than those obtained from the local calculations [25], and the
latter agree quite well with the flowing afterglow-Langmuir
probe (FALP) experiment [25]. On the other hand, our previous
nonlocal calculations of rate coefficients for Cl2 [8] and F2 [9]
also agree very well with FALP measurements.
To investigate the problem further, we performed local
and nonlocal calculations of DEA to F2 through the u
resonance using potential curves and fixed-nuclei K matrices
[37] calculated by the same ab initio methods as in Ref. [25].
Since the DEA process at low energies is dominated by the
p wave in this case, we included only two terms, l = 1 and
3 in the partial-wave expansion, and did not analyze the angular distribution. The total nonlocal width was parametrized
according to Eq. (16) with the threshold exponent a = 3/2.
In Table III we present the fit parameters A, B, and the total
local width loc for six internuclear separations. As in the case
of ClF, the parameters were obtained from the fit to the ab
initio K matrices [37].
The results for DEA cross sections are presented in Fig. 8,
where we also compare with previous resonance R-matrix
calculations [9] in which the R-matrix parameters were fixed
by adjusting the resonance width to the results of Hazi
et al. [38]. The resonance R-matrix theory is completely
equivalent to the nonlocal theory [39]; therefore, the difference
between the results of two calculations should be attributed
TABLE III. F2 : parameters for the nonlocal width entering
Eq. (16), and the local width loc ; all quantities are in a.u.
R
2.40
2.45
2.50
2.55
2.60
2.65
052707-6
B
A
loc
0.048 82
0.044 34
0.037 37
0.029 13
0.019 57
0.008 05
0.1790
0.1553
0.1236
0.0884
0.0507
0.0141
0.065 64
0.049 27
0.033 53
0.019 82
0.009 06
0.001 89
DISSOCIATIVE ELECTRON ATTACHMENT TO HALOGEN . . .
PHYSICAL REVIEW A 94, 052707 (2016)
FIG. 8. DEA cross sections for the F2 molecule in the ground
vibrational state. Solid (black) line: present nonlocal theory; solid
(red) line RM: resonance R-matrix calculations [9] with the resonance
width adjusted to Hazi et al [38]; dashed (black) line nB and solid
(blue) line B: local theory without and with Bardsley correction,
respectively.
FIG. 9. Survival factor of the local theory for F2 and ClF. Solid
(black) curves: exact result from Eq. (20). Dashed (red) curves: WKB
approximation, Eq. (22).
to different resonance width. On the other hand, the local
cross sections are very different from both calculations. At
E < 0.4 eV the local theory results are substantially lower than
nonlocal, except the low-energy region below 0.03 eV where
the calculations without Bardsley correction exhibit the 1/E
divergence. Similar results for DEA to the F2 molecule were
obtained by Houfek et al. [40]: their local cross sections for the
“F2 -like model” are substantially smaller than nonlocal in the
energy interval between 0.03 and 0.6 eV. It is also interesting
that the exact cross sections obtained within the framework of
their model are close to the results of the nonlocal theory whose
only assumption is the Born-Oppenheimer approximation.
In addition, our local theory results exhibit an oscillatory
structure. These oscillations are not unexpected, and might
appear [41] due to the oscillatory energy dependence of the
survival factor defined as [42]
s = |S|,
(20)
where S is the nonunitary scattering matrix obtained from the
solution of Eq. (9) with the zero right-hand side. To understand
better the structure of the local cross section it is useful to
analyze it from the point of view of the quasiclassical (WKB)
theory. The quasiclassical approximation leads to the representation of the DEA cross section as a product of the capture
cross section and the survival factor. However, expressions
for these quantities depend on a specific version of the WKB
expansion. In the original theories of Bardsley, Herzenberg,
and Mandl [43] and O’Malley [42] the expansion of the WKB
wave function is carried out about the classical turning point
for the anion motion. A more rigorous quasiclassical theory
is based on the saddle-point technique with the uniform Airy
function approximation [33]. Application of this method leads
to the FC transition point Rv defined by Eq. (13). Although
the FC point is complex, we will neglect again its imaginary
part that introduces a small error for narrow resonances. This
approximation is traditional in WKB versions of the local
theory [42,43]. The result for the DEA cross section in this
approximation can be written as
σ =
π2
(Rv )Fv (E)s,
E
(21)
where Fv (E) is the generalized FC factor for which a
uniform Airy function approximation has been worked out
by Kazansky and Yelets [33]. The survival factor in the WKB
approximation is well known [32,43]. Assuming also a small
resonance width, we can simplify it further as
Rs
(R)
dR ,
(22)
s = exp −
a− v(R)
where v(R) is the classical velocity for the anion motion and
a− is the turning point for the anion motion. We neglect again
the imaginary part of a− .
In Fig. 9 we compare the exact survival factor, Eq. (20), with
the approximate survival factor calculated from Eq. (22). It is
apparent that the oscillations in the exact survival factor lead
to the oscillations in the DEA cross sections. It is interesting,
though, that for heavier targets these oscillations disappear. In
the same figure we show comparison of survival factors for
ClF. No oscillations are seen in the exact survival factor for
ClF. At zero electron energy the approximate survival factor
is equal to 1, although the exact survival factor is somewhat
smaller due to contribution of internuclear distances left of the
transition point.
In Fig. 10 we compare the WKB version of the local theory
with nonlocal calculations. Although both versions of the local
theory are substantially different from the nonlocal results,
in the low-energy region the WKB version produces much
better results than the exact local theory. As can be seen
from Eq. (21), the threshold behavior is determined by the
ratio (Rv )/E. When E → 0 the FC point Rv approaches the
052707-7
I. I. FABRIKANT
PHYSICAL REVIEW A 94, 052707 (2016)
FIG. 10. DEA cross section for F2 , comparison of the present
nonlocal results, solid (black) curve, with the WKB version of the
local theory, dashed (red) curve.
crossing (stabilization) point Rs ; therefore, if for R → Rs the
width is parametrized as
(R) = const × (U (R) − V0 (R))a ,
then according to Eq. (13)
(Rv ) = const × E a ,
where a is the correct threshold exponent (3/2 in case of F2 ),
and the cross section exhibits the correct low-energy behavior
as shown in Fig. 10.
We conclude that the WKB version of the local theory works
better than the exact local theory itself. A similar conclusion
can be drawn from comparison of the two versions of the
local theory for ClF. Although a complete analysis of this
interesting phenomenon is beyond the scope of the present
paper, we present here two important points.
(1) Since there is one-to-one correspondence between the
electron energy and the transition point in the WKB version, it
is easier to enforce the correct threshold behavior in the WKB
version as discussed above, whereas the Bardsley correction of
the exact local theory might strongly underestimate the DEA
cross section.
(2) In both theories involving the WKB approximation
(local and nonlocal) we also use the FC approximation. It
might be argued that this can cause a similarity between our
nonlocal calculation and the WKB version of the local theory,
just reflecting the error of the WKB and FC approximations.
However, the fact that our previous R-matrix calculations for
F2 [9], based on the WKB approximation, agree with the
quantum-mechanical calculations of Hazi et al. [38], indicates
that the WKB approximation and FC principle work very well
in this system.
This bring us back to the point of comparison between
the local theory, nonlocal theory, and experiment. As can be
seen from Fig. 8, our previous calculations for F2 , based on
the data of Hazi et al. [38] for the resonance width, in the
low-energy region are about a factor of 2 lower than the present
calculations. The analysis of the corresponding resonance
width leads to the same conclusion: the present resonance
width is about a factor of 2 higher than one calculated in
Ref. [38]. Since at low electron energies the survival factor
is close to 1, the cross section is roughly proportional to the
resonance width at the FC point, and this explains the factor of
2 difference between the present cross sections at low energies
and those calculated in Ref. [9]. The same argument can be
presented for the thermal rate coefficient.
This is helpful for analysis of our results for ClF. A reason
for disagreement of the thermal rate coefficients calculated
from the nonlocal cross section with experimental rate coefficients might be an overestimation of the width by a factor of
2 in the region of internuclear distances close to the crossing
point. On the other hand, the local theory, for a given width, underestimates the thermal rate coefficients. Therefore, our local
results [25] agree well with experiments. It should be stressed,
however, that due to the sensitivity of the DEA cross sections to
all aspects of the theory, including the method of calculation of
fixed-nuclei parameters and the nuclear dynamics, the factor of
2 accuracy should be considered as good. We also stress that at
low energies the DEA cross section is extremely sensitive to the
behavior of the width near the crossing point. Additional nonlocal calculations for ClF have shown that the thermal rate coefficient can be easily reduced by a factor of 2 by reducing the
width only at R 3 a.u. (with the crossing point Rs = 3.02).
This is the range of internuclear distances where the accurate
calculation of the resonance width is particularly very challenging. As was shown in Ref. [44], different methods of calculation of the resonance width for F2 give very different results
which sometimes differ by the order of magnitude. (We should
note, though, that the results of Hazi et al. [38] for the width are
plotted in Ref. [44] incorrectly). Because of such difficulties in
accurate calculation of (R), DEA experiments can actually
serve as a method of calibration of this function. In the case of
F2 it appears that the 35-year-old results of Hazi et al. for the
width are the most accurate for the purpose of calculation of
thermal DEA rate coefficients as demonstrated in Ref. [9].
IV. CONCLUSION
The presented results confirm once again that calculation
of DEA cross section is a very nontrivial task, even if all
fixed-nuclei input parameters are known. Although the local
version of the DEA theory turned out to be successful for
several nonpolar molecules, the halogen molecules represent
another example, in addition to hydrogen halides, where
nonlocal calculations are necessary for an accurate description
of cross sections and thermal rate coefficients. This is mainly
due to a special behavior of the neutral and anion curves in the
Franck-Condon region: they cross very close to the equilibrium
internuclear separation for the neutral. Whereas at very low
energies the local theory overestimates DEA cross sections
due to the 1/E divergence, at higher-energy region, important
for calculation of thermal rate coefficients, the local results
are lower than nonlocal. Investigation of the quasiclassical
version of the local theory is helpful in understanding the
difference between the local and nonlocal results.
Our results for differential cross sections for DEA to
ClF show the dominance of the p-wave contribution to
052707-8
DISSOCIATIVE ELECTRON ATTACHMENT TO HALOGEN . . .
the DEA process above E = 0.01 eV. Below this energy
a graduate transition from the p-wave dominance to the
s-wave dominance occur. This tells us that in spite of the
inversion symmetry breaking in ClF, the DEA process in
this interhalogen molecule is similar to that in homonuclear
halogens F2 and Cl2 .
[1] W.-C. Tam and S. F. Wong, J. Chem. Phys. 68, 5626 (1978).
[2] L. G. Christophorou and J. K. Olthoff, J. Phys. Chem. Ref. Data
28, 131 (1999).
[3] E. P. Wigner, Phys. Rev. 73, 1002 (1948).
[4] D. Klar, M.-W. Ruf, and H. Hotop, Chem. Phys. Lett. 189, 448
(1992); Aust. J. Phys. 45, 263 (1992).
[5] S. Barsotti, M.-W. Ruf, and H. Hotop, Phys. Rev. Lett. 89,
083201 (2002).
[6] M.-W. Ruf, S. Barsotti, M. Braun, H. Hotop, and I. I. Fabrikant,
J. Phys. B 37, 41 (2004).
[7] M. Braun, M.-W. Ruf, I. I. Fabrikant, and H. Hotop, Phys. Rev.
Lett. 99, 253202 (2007).
[8] J. F. Friedman, T. M. Miller, L. C. Schaffer, A. A. Viggiano, and
I. I. Fabrikant, Phys. Rev. A 79, 032707 (2009).
[9] N. S. Shuman, T. M. Miller, A. A. Viggiano, and I. I. Fabrikant,
Phys. Rev. A 88, 062708 (2013).
[10] W. Domcke, Phys. Rep. 208, 97 (1991).
[11] I. I. Fabrikant, Phys. Rev. A 43, 3478 (1991).
[12] C. Mündel, M. Berman, and W. Domcke, Phys. Rev. A 32, 181
(1985).
[13] M. Čižek, J. Horáček, and W. Domcke, Phys. Rev. A 60, 2873
(1999).
[14] M. Čižek, J. Horáček, A.-Ch. Sergenton, D. B. Popović, M.
Allan, W. Domcke, T. Leininger, and F. X. Gadea, Phys. Rev. A
63, 062710 (2001).
[15] M. Čižek, J. Horáček, M. Allan, I. I. Fabrikant, and W. Domcke,
J. Phys. B 36, 2837 (2003).
[16] I. I. Fabrikant, J. Phys. Conf. Ser. 192, 012002 (2009).
[17] S. T. Chourou and A. E. Orel, Phys. Rev. A 77, 042709 (2008).
[18] D. T. Birtwistle and A. Herzenberg, J. Phys. B 4, 53 (1971).
[19] A. U. Hazi, Phys. Rev. A 19, 920 (1979).
[20] H. Feshbach, Ann. Phys. (N.Y.) 5, 357 (1958).
[21] A. M. Lane and R. G. Thomas, Rev. Mod. Phys. 30, 257 (1958).
[22] J. Macek, Phys. Rev. A 2, 1101 (1970).
[23] D. J. Haxton, C. W. McCurdy, and T. N. Rescigno, Phys. Rev.
A 73, 062724 (2006).
PHYSICAL REVIEW A 94, 052707 (2016)
ACKNOWLEDGMENTS
The author thanks V. Kokoouline for providing his unpublished results for e−F2 fixed-nuclei scattering matrices. This
work was supported by the US National Science Foundation
under Grant No. PHY-1401788.
[24] J. R. Taylor, The Quantum Theory of Nonrelativistic Collisions
(Dover, Mineola, NY, 2000).
[25] J. P. Wiens, J. C. Sawyer, T. M. Miller, N. S. Shuman, A. A.
Viggiano, M. Khamesian, V. Kokoouline, and I. I. Fabrikant,
Phys. Rev. A 93, 032706 (2016).
[26] J. Tennyson, D. B. Brown, J. J. Munro, I. Rozum, H. N.
Varambhia, and N. Vinci, J. Phys. Conf. Ser. 86, 012001
(2007).
[27] J. Tennyson, Phys. Rep. 491, 29 (2010).
[28] J. M. Carr, P. G. Galiatsatos, J. D. Gorfinkiel, A. G. Harvey,
M. A. Lysaght, D. Madden, Z. Mašı́n, M. Plummer, J. Tennyson,
and H. N. Varambhia, Eur. Phys. J. D 66, 58 (2012).
[29] L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Nonrelativistic theory) (Pergamon Press, New York, 1965).
[30] T. F. O’Malley and H. S. Taylor, Phys. Rev. 176, 207 (1968).
[31] N. F. Lane, Rev. Mod. Phys. 52, 29 (1980).
[32] J. N. Bardsley, J. Phys. B 1, 349 (1968).
[33] A. K. Kazansky and I. S. Yelets, J. Phys. B 17, 4767 (1984).
[34] J. N. Bardsley, in Electron-Molecule and Photon-Molecule
Collisions, edited by T. Rescigno, V. McKoy, and B. Schneider
(Plenum, New York, 1979).
[35] S. A. Kalin and A. K. Kazansky, J. Phys. B 23, 4377 (1990).
[36] K. Gope, V. S. Prabhudesai, N. J. Mason, and E. Krishnakumar,
J. Phys. B 49, 015201 (2016).
[37] V. Kokoouline (unpublished).
[38] A. U. Hazi, A. E. Orel, and T. N. Rescigno, Phys. Rev. Lett. 46,
918 (1981).
[39] I. I. Fabrikant, Comments At. Mol. Phys. 24, 37 (1990).
[40] K. Houfek, T. N. Rescigno, and C. W. McCurdy, Phys. Rev. A
77, 012710 (2008).
[41] G. A. Gallup, J. Chem. Phys. 126, 064101 (2007).
[42] T. F. O’Malley, Phys. Rev. 150, 14 (1966).
[43] J. N. Bardsley, A. Herzenberg, and F. Mandl, Proc. R. Soc. 89,
321 (1966).
[44] M. Honigmann, R. J. Buenker, and H. P. Liebermann, J. Comput.
Chem. 33, 355 (2012).
052707-9